We apply concepts of covariant and contravariant vector space in differential geometry and general relativity to derive new, general, exact relations between potential of mean force and free-energy profile. These relations are immensely practical in free-energy simulations because a full Jacobian transformation (which is usually unknown) is not required; rather, only knowledge of the (constraint) coordinate of interest is needed. We reveal that in addition to the Jacobian determinant, the Jacobian scale factor and Leibnizian contributions must also be considered, as well as a Fixman term with correct mass dependence. Our newly derived relations are verified with new nontrivial benchmark numerical examples for which exact results can be computed and compared with relations available in the literature that turn out to exhibit significant deviations from the exact values.
